Fuzzy Structures on Z Algebras

Abstract

In 1966, Imai and Iseki [25,26] introduced two new classes of abstract algebras: BCK-algebras and BCI-algebras. These algebras have been extensively studied since their introduction. In 2017, Chandramouleeswaran et al.[18] introduced the concept of Z-algebras as a new structure of algebra based on propositional calculus. The Z-algebra is not a generalization of BCK/BCI-algebras. In 1965, Zadeh[73] introduced the fundamental concept of a fuzzy set which is a generalization of an ordinary set. The fuzzy set theories developed by Zadeh and others are found many applications in the domain of mathematics and elsewhere. In 1971, Rosenfeld [65] introduced the notion of fuzzy groups. In 1991, following the idea of fuzzy groups, Xi[71] introduced the notion of fuzzy BCK-algebras. In 1994, Jun and Meng [32] introduced the notion of fuzzy p-ideals and in 1999, Khalid and Ahmad [40] introduced the concept of fuzzy H-ideals in BCI-algebras and studied their properties. In 1997, Meng et al. [49] and Mostafa [50] fuzzified the concept of implicative ideals in BCK-algebras, independently. In the year 2009, fuzzy translations and fuzzy multiplications in BCK/BCI-algebras have been discussed by Lee et al.[44]. newlineIn 1986, the idea of intuitionistic fuzzy set was first published by Atanassov [8], as a generalization of the notion of fuzzy set. In 1984, intuitionistic L-fuzzy set was introduced by Atanassov and Stoeva [11] as a generalization of L-fuzzy set. In 1975, Zadeh [74] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets [73]. In 1989, K. T. Atanassov and G. Gargov [10] proposed interval-valued intuitionistic fuzzy set based on the comparative analysis of interval-valued fuzzy sets and intuitionistic fuzzy sets. Intuitively, the extension of intuitionistic fuzzy sets to interval-valued intuitionistic fuzzy sets furnishes additional capability to handle the vague information. In 2012, Jun et al. [36] have introduced a remarkable theory, namely, the theory of cubic sets. This structure is comprised